Independence proof concerning compact cardinals, continuum hypotbesis, and normal ultrafilters

关于紧基数、连续统假设和普通超滤器的独立性证明

• 批准号: 09640299
• 负责人: ABE Yoshihiro
• 金额: $ 0.9万
• 依托单位: Kanagawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640299/
• 关键词: compact cardinal; continuum hypothesis; normal ultrafilter; forcing; indescribability; bounded ideal; partition property; precipitous; Precipitous; compact cardinal; seminormal ideal; clused unbounded filter; diamond principle; very weak squa

We investigated a K<@D1+@>D1-C.C.(]SY.di-substituted right.[)K-strategically closed forcing adding arbitrary nunber of closed unbounded subsets of K.A Lottery preparation followed by this forcing preserves the strong compactness of K.In the resulting model the restricions of all normal ultrafilters to the ground model coincide.We know a symmetric model in which the genemlized continuum hypothesis holds below a strongly compact cardinal K and fails at K.However it turned out establishing the same thing together with the axiom of choice is far more difficult than had been expected. It is not clear whether the strong compactness of K is preserved when we force the the axion of choice on the symmetric model.Several facts and new technique are founded as in the folllowing.Forcing : (1)Adding many closed unbounded sets (2)Adding non-reflecting stationary sets to P_klambda (3)Forcing a non-regular ultrafilter on P_kK^<+++> with K supercompact.Combinatorics : (1)Combinatorial characterization of PIi-indescribability in P_klambda.(2)Bounded ideal may or may not have the partition property.(3)Bounded ideal is not precipitousIt is expected that further research makes the relation between the partition property and ineffability of P_klambda clear to bring great progress in the combinatorics of P_klambda.

我们调查了k <@d1+@> d1-c.c。（] sy.di取代的权利。[）k战略性地闭合迫使添加了任意封闭的无限制子集的k..a彩票制备子集的nunber，然后强迫保留了k的强大压实性。 Continuum假设保持在强烈紧凑的基础K以下，而在K处失败。但是，事实证明，与选择的公理建立相同的东西远比预期的要困难得多。目前尚不清楚当我们强迫对称模型上的首选轴承时是否会保留k的强紧凑性。几个事实和新技术的建立如在卵泡中一样建立。（1）福音：（1）添加许多封闭的无界套装（2）添加非反射的固定集（2）添加p_klambda（3）的非反射式（3）<+ker+ker+++ker+++kn++kn++kn+++kn++++prof^++prof^++prof^< supercompact.combinatorics：（1）P_KLAMBDA中PII可视性的组合表征。（2）有限的理想可能或可能没有分区属性。（3）有界的理想并不是悬而未决的，预计进一步的研究使得对P_Klambda的分区属性和无能为力之间的关系使P_Klambda的差异很大。

项目成果

Yoshihiro Abe: "A variant of the diamond principle for combinatorial ideals" Proceedings of the American Mathematical Society. 127-3. 847-849 (1999)

Yoshihiro Abe：“组合理想的钻石原理的变体”美国数学会论文集。

Yoshihiro Abe: "A variant of the diamond principle for combinatorial ideals" Proceedings of the American Mathematical society. (1998)

安倍佳弘：“组合理想的钻石原理的变体”美国数学会论文集。

Sakae Fuchino and Lajos Soukup: "More set-theory around the Weak Freese-Nation property" Fundamenta Mathematicae. 154-2. 159-176 (1997)

Sakae Fuchino 和 Lajos Soukup：“有关弱自由国家性质的更多集合论”基础数学。

Yo Matsubara and Masahiro Shioya: "Nowbere precipitousness of some ideals" Journal of Symbolic Logic. 63-2. 1003-1006 (1998)

Yo Matsubara 和 Masahiro Shioya：“现在有些理想的陡峭性”《符号逻辑杂志》。

Yoshihiro Abe: "A hierarchy of filters smaller than CF_<Kλ>" Archive for Mathematical Logic. 36・6. 385-397 (1997)

Yoshihiro Abe：“小于 CF_<Kλ> 的滤波器层次”数学逻辑档案 36・6（1997）。